Why does angular velocity changes as i strech my hands?

Question

Suppose I am sitting on a turnable chair. I have given that turnable chair some angular velocity $\omega$ and it starts rotating around me as axis of rotation.

Now at that instant suppose I expand out my hands with some weight on both sides.Now, It's mentioned in my book that the angular velocity decreases!! i.e $\omega'\lt \omega$.

Again as I brings my hands closer my angular velocity is restored.

I was wondering what's the possible reason behind this?

My question:- Why the angular velocity decreases with expansion of my hands and why its restored again?

Moreover what is the role of conservation of angular momentum in this case and how?

Answer 1

Your angular velocity changes depending on your arms' extension due to conservation of angular momentum.

Conservation of angular momentum implies that the quantity $$\bf \vec L=I{\bf\vec\omega}={\text{constant}}$$ This means in a rotating system changing from some state $1$ to another $2$ then $$\bf I_1\omega_1=I_2\omega_2=\text{constant}$$

Applying this to your situation, if your moment of inertia (see image of figure skater here) is initially $I_1$ when your hands are closer together, and you also have angular velocity $\bf\omega_1$ then when you extend your hands you have a new moment of inertia $I_2$ where$^1$ $I_2\gt I_1$ and so from the above equation $\bf\omega_1\gt\omega_2$ where $\bf\omega_2$ is your new angular velocity.

So your angular velocity is smaller when you extend your arms, and higher when you bring them closer so that angular momentum stays the same.

Moreover what is the role of conservation of angular momentum in this case and how?

Conservation of angular momentum is playing a fundamental role here as explained above.

$^1$ Moment of inertia is larger when more mass is distributed further from the axis of rotation.